The above example shows that it has been long known that the law of total probability cannot be taken for granted in quantum theory. Indeed, many other classical probability rules are only upheld in specific conditions. For instance, a joint probability can be definitely assigned only when the two measurement operators are commutative [1]}, [2]}, [3]}, [4]}; There are many variants of definition of the conditional probability in quantum theory (see a review in [5]}). However, a family of no-go theorems recently published[6]}, [7]}, [8]}, [9]} appear to rely on the total law of probability one way or another without considering the sufficient conditions. These no-go theorems are related to the extensively discussed Wigner's Friend experiments. In quantum mechanics, the Wigner's friend [10]}, [11]} thought experiment has been widely discussed as it tests the validity of many quantum interpretation theories. The significance of such experiment is that Wigner and his friend give two different descriptions of the same physical process happened inside the lab. Deutsch further extended the thought experiment to be applicable to macroscopic system such as the lab system [12]} itself. Based on that, a more sophisticate extended Wigner's friend experiment is put forwarded by Brukner [13]}, [6]}. Such experiment setup involves two remotely separated labs. Each lab contains half of an entangled pair of spins and a local observer. Outside each lab there is a super-observer who can choose to perform different types of measurements on the lab as a whole. The intention of such experiment setup is to prove, through a no-go theorem, that measured facts are observer-dependent in quantum theory. Subsequent experiment [15]} has been carried out to confirm the inequality developed in [6]}. A stronger version of no-go theorem is further proposed for reaching the similar conclusion [7]}. The statement that measured facts are observer-dependent is considered of importance for the quantum foundation and deserved rigorous theoretical proving and experimental testing. However, proving the no-go theorems by taking the law of total probability for granted casts doubt on the theoretical rigorousness.
